Integration by Substitution (Section 7.2): Study Notes

Integration by Substitution

Introduction to Substitution

Integration by substitution is a fundamental technique in calculus, analogous to the Chain Rule for derivatives. It is used to simplify integrals by transforming them into a more manageable form. This method is especially useful when an integral contains a composite function.

• Chain Rule for Derivatives: If $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$.

• Integration by Substitution: Reverses the chain rule to integrate composite functions.

Substitution Method

Steps for Integration by Substitution

To apply the substitution method, follow these steps:

1. Let $u$ be a function of $x$: Choose $u = g(x)$ such that its derivative $g'(x)$ also appears in the integrand.

2. Rewrite the Integral: Express the entire integral in terms of $u$ and $du$.

3. Integrate with Respect to $u$: Perform the integration in the new variable.

4. Substitute Back: Replace $u$ with the original expression in $x$.

In general, for the types of problems we are concerned with, there are four cases:

• The integrand is a product of a function and its derivative.

• The integrand contains a composite function whose inner function's derivative is present.

• The substitution simplifies the integral into a basic form.

• The substitution is chosen to match a standard integral formula.

Common Substitution Integrals

The following table summarizes common integrals and their substitutions:

Form of the Integrand	Substitution	Result
$\int f'(g(x)) \cdot g'(x)\,dx$	$u = g(x)$	$\int f'(u)\,du = f(u) + C$
$\int x \cos(x^2)\,dx$	$u = x^2$	$\frac{1}{2}\sin(x^2) + C$
$\int \frac{1}{x}\,dx$	$u = x$	$\ln|x| + C$
$\int e^{3x}\,dx$	$u = 3x$	$\frac{1}{3}e^{3x} + C$

Worked Examples

Example 1: $\int 2x \cos(x^2)\,dx$

• Substitution: Let $u = x^2$, so $du = 2x\,dx$.

• Rewrite: $\int \cos(u)\,du$

• Integrate: $\sin(u) + C$

• Back Substitute: $\sin(x^2) + C$

Example 2: $\int (4x + 13)^5\,dx$

• Substitution: Let $u = 4x + 13$, so $du = 4\,dx$ or $dx = \frac{du}{4}$.

• Rewrite: $\int u^5 \cdot \frac{du}{4}$

• Integrate: $\frac{1}{4} \cdot \frac{u^6}{6} + C = \frac{1}{24}u^6 + C$

• Back Substitute: $\frac{1}{24}(4x + 13)^6 + C$

Example 3: $\int x e^{x^2}\,dx$

• Substitution: Let $u = x^2$, so $du = 2x\,dx$ or $x\,dx = \frac{du}{2}$.

• Rewrite: $\int e^{u} \cdot \frac{du}{2}$

• Integrate: $\frac{1}{2}e^{u} + C$

• Back Substitute: $\frac{1}{2}e^{x^2} + C$

Example 4: $\int \frac{1}{x}\,dx$

• Integrate directly: $\ln|x| + C$

Application Example: Business Context

Debt Accumulation Problem

A company incurs debt at a rate of $D'(t) = 1000 + 0.1t^2$ (dollars per year), where $t$ is the number of years since borrowing began. The initial debt is $D(0) = 5000$ dollars.

• Find the total debt after $t$ years:

Integrate the rate function:

$D(t) = \int (1000 + 0.1t^2)\,dt + 5000$

$= 1000t + 0.1 \cdot \frac{t^3}{3} + 5000$

$= 1000t + \frac{0.1}{3}t^3 + 5000$

• Find when the debt exceeds $34,000$ dollars:

Set $D(t) = 34,000$ and solve for $t$:

$34,000 = 1000t + \frac{0.1}{3}t^3 + 5000$

$29,000 = 1000t + \frac{0.1}{3}t^3$

This equation can be solved numerically for $t$.

Summary Table: Substitution Integrals

Integral	Substitution	Result
$\int 2x \cos(x^2)\,dx$	$u = x^2$	$\sin(x^2) + C$
$\int (4x + 13)^5\,dx$	$u = 4x + 13$	$\frac{1}{24}(4x + 13)^6 + C$
$\int x e^{x^2}\,dx$	$u = x^2$	$\frac{1}{2}e^{x^2} + C$
$\int \frac{1}{x}\,dx$	$u = x$	$\ln|x| + C$

Key Points to Remember

• Integration by substitution is the reverse process of the chain rule for derivatives.

• Choose a substitution that simplifies the integral, typically where the derivative of the inner function is present.

• Always substitute back to the original variable after integrating.

• This method is widely used in business calculus for solving accumulation and growth problems.